Appendix I. Subharmonic and Plurisubharmonic Functions
Subharmonic functions and potential theory are often used in the theory of one complex variable. For holomorphic functions of several complex vari• ables defined in a domain Q c Definition 1.1. Let QcJRm be a domain. A real valued function cp(x) with values in [ - 00, + (0) is said to be subharmonic in Q if i) cp(x) is upper semi-continuous and cp(x) $ - 00; ii) for every XEQ and every r where dW m is the Lebesgue measure on the unit sphere sm-l and Wm is the total mass of sm-l. We denote by SeQ) the family of subharmonic functions in Q. If cp and - cp are subharmonic in Q, we say that cp is harmonic in Q. Remark. If cp is subharmonic in Q and l' < dQ(x), then cp(X):;:;,,;;l r- 2m S cp(x+xl)d'm(x/)=A(x, r, cp) Ilxll ;:;;r where dIm is the Lebesgue measure in JRm and 'm is the total mass of the unit ball B(O, 1). Definition 1.2. Let Q c 2" cp(z);;;;(2n)-1 S cp(z+rei8 w)dO. o Appendix I. Subharmonic and Plurisubharmonic Functions 231 If q> and - q> are plurisubharmonic, we say that q> is pluriharmonic in Q. We denote by PSH(Q) the family of functions plurisubharmonic in Q. Remark 1. If Qc(Cn, then PSH(Q)cS(Q) and if n=1, PSH(Q)=S(Q). Remark 2. q> c PSH (Q) if and only if it is upper semi-continuous, not identi• cally - 00, and if its restriction to every complex line Ll meeting Q is either subharmonic or the constant - 00 on each connected If-open set of Ll "Q. Examples. 1) Any continuous convex function in Q (in terms of the underly• ing real variables) is in PSH(Q), for then we have q>(x) ~H Proposition 1.3. i) If q>E PSH(Q) and c > 0, then Cq>E PSH(Q). ii) If q>l and q>2 are in PSH(Q), then sup (q>l' q>2)EPSH(Q). iii) rr q>v is a decreasing sequence of plurisubharmonic functions in Q, then either lim q>v(z)== - 00 or q>(Z) = lim q>v(z)EPSH(Q). Proof These are immediate consequences of Definition 1.2. o Definition 1.4. A function q>ES(Q) will be said to be continuous if it is continuous for the completion of the Euclidean topology on' IR to the point -00. Remark. q>ES(Q) is continuous if and only if exp q>(x) is continuous for the Euclidean topology on IR. PropositionI.5. If QcIR»! and q>E~2(Q), then q>ES(Q) if and only if m ij2 L1 q>(x)~O, where L1 is the Laplacian L1 = i~l ox;- If Qc (Cn and q>E~2(Q) then 232 Appendix 1. Subharmonic and Plnrisubharmonic Functions Proof We write the Taylor series expansion of Then since A(x,r,xj-x)=O for all j by symmetry (this is an odd function) and ).(x,r,(xj-x)(X~-Xk))=O forjtok, again by symmetry. Th~ 1 lim [A(X, r, On the other hand, if Will is the measure of the unit sphere sm-I in ]Rill, w",=2n"'i 2 [F(m/2)]--I, then we obtain from Gauss' Theorem (or Green's Theorem) ,. (I,1 ) A(x,r, for every WE Proposition 1.6. If Qe]Rm and Proof It follows from (I,1) that 8), 8},(x, 1', If Qden and qJEPSH(Q)n~2(Q), then it follows from (1,1) for m=2 and (1,2) that a 2" . v(z, z', r)=-a-l- S qJ(z+re'Oz')dO ogr 0 is increasing in r. Since this is true for all z', v(z, r) = W Zn1 S v(z, z', r)dw2n (z') =~ Jc(z, r, qJ) Ilz'II=1" ologl is an increasing function of log r, hence },(z, r, qJ) is convex in log r. 0 Remark. The result that Jc(z, r, qJ) is increasing and convex in logr is an important property of plurisubharmonic functions and is not in general true for IR. 2n-subharmonic functions. In the definition of subharmonic and plurisubharmonic functions, we require only upper semi-continuity, whereas in Propositions 1.5 and 1.6, we made assumptions on the regularity of the functions. We now extend these properties in a way to drop the regularity assumptions. Lemma 1.7. Let Q c IR. m be a domain and 0 < c ~ 1. Suppose that Q' c Q and for XEQ', B(x,cdQ(x))cQ'. Then either Q'=Q or Q'=0. Proof By the hypothesis, Q' is open. Suppose that xoEQ' n Q and let d=dQ(xo)' Then there exists a point X'EQ' such that Ilx' -xoll ~cd/4. This implies that dQ(x')~3d/4 and xoEB(x', cdQ(x'))cQ'. Thus Q' is also closed, and since Q is connected, Q' = Q or Q' = 0. 0 In Dz,w= {Z'E Lemma 1.8. Let Qc Proof Obviously, S(z, Q) is a disked neighborhood of z. Let ZoES(Z, Q) be such that Zo =l= z. Then Zo = z + z' with z' =l= 0 and the disc. D z •z ' is compact in Q. There exists a disked open neighborhood U of the origin such that Dzoz' +UcQ. But Dz,z'+U is a union of discs centered at z, so Dz,z' + U c S(z, Q). Thus S(z, Q) contains an open neighborhood of Zo and hence IS open. To prove the second part of the lemma, we note that S(z, Q) contains the ball B(z, dQ(z)) and so the conclusion follows from Lemma 1.7. 0 Proposition 1.9. If Qc Proof Let N be the set of points in Q such that S cpdT Zn = - 00 for every u neighborhood U of z,U~Q. For zEN and [[z'-z[[ Corollary 1.10. For Q a domain in IR m (respectively Proof For CPl' cpzEPSH(Q) (or S(Q)), the set {z: CPl(Z)= - 00 or cpz(z) = - oo} is of measure zero by Proposition 1.9. Thus, tCPl +(1-t)cpz is not identically - 00, O:;:;t:;:; 1, and hence is in PSH(Q) or S(Q). D Definition 1.11. A subset E c Q, a domain in IR'" (resp. Corollary 1.12. A (pluri)polar set in a domain Q c Proposition 1.13 (Maximum Principle). Let QcIR'" be a domain and cpcS(Q). Let m=sup cp. If there exists xoEQ such that cp(xo)=m, then cp=m. Q Proof If B(xo,r)cQ, then m=cp(xo):;:;A(x,r,cp):;:;m. Thus, cp(x)=m in B(xo, r), for otherwise, by the upper semi-continuity of cP there would exist 8>0 and an open subset U of B(xo, r) for which cp(x) PropositionI.14. Let cp(z,t) be a real valued function of ZEQc Proof We shall verify properties i) and ii) of Definition 1.2. Let l/J*(z) = lim sup l/J(z) be the upper regularization of l/J(z). z'-tz Then there exists a sequence W q such that W q -+ 0 and l/J*(z) = lim l/J(z+ wq)= lim sup S cp(z+Wq, t)d/1(t). q- 00 q-HX) Appendix 1. Subharmonic and Plurisubharmonic Functions 235 From Fatou's Lemma and the uniform bound for tjJ*(z)~S lim sup q~Cf) where the second inequality stems froms the semi-continuity of Remark. Proposition I.14 remains valid for Proposition 1.15. Let i) Proof By ii) of Definitions I.1 and 1.2 we obtain CPt(x)~cp(x) for 8 CPJX) = S cp(x+ y)rxe(y)dr(v) = S cp(x+8y')rx(y')dr(y')~ From this we obtain for s' < s Subharmonic and plusubharmonic functions are locally integrable. Thus, using differentiation for distributions, we extend to SeQ) and so PSH (Q) the properties given first for differentiable subharmonic and plurisubharmonic functions. For and for (1,3) (1,3) is a distribution in Q depending on the vector w. Proposition 1.16. Let Proof Let tjJEee;::(Q), tjJ;=;;O. By Proposition 1.15, there exists a sequence q~ co sure. Similarly for L( Proposition 1.17. Let (pEPSH(Q). Then Jc(z, r, M Proof For 11 >0, let (P q be a sequence of ee co plurisubharmonic functions such that M",(z', r) is a plurisubharmonic function of the variable z" for fixed z', and M",(z',r)=Az"(O,r,M,,,(z',r)) is an increasing convex function oflogr. 0 Definition 1.18. A function Proposition 1.19. If Proof Let - 00 such that }, (~, r, 0. Furthermore, Corollary 1.20. Let Q be a domain in domains Ui (i.e. Q = i9i Ui ). If CorollaryI.21. Let MeQ be closed and Me{x: Proof Suppose not. Then we have a decomposition Q = Qi U Q2 U M with Qi nQ2 =0. It follows from Proposition I.9 that M=0. Since M nQi =MnQ2=0, there exists ~EM such that ~EbdQinbdQ2' Let r be chosen such that r < dQ(~)' Then B(~, r/2) n Qi =1= 0 and B(~, r/2) n Q2 =1= 0. Let q~oo 238 Appendix I. Subharmonic and Plurisubharmonic Functions g*(x)=limsupg(x') is subharmonic. But g*(~)=0>A(~,r/2,g*), which is a x'--+X contradiction. If M is an analytic variety, take r Proposition 1.22. Let Q c lR m be a domain (resp. Q c Proof For ~EM and r Corollary 1.23. Suppose that Q c Proof Let f(z)=u(z)+iv(z). We apply first Proposition I.22 to u(z) and - u(z), which are subharmonic in Q n CM. Thul' , there are subharmonic functions in Q, u(z) and - u(z) which extend u(z) and - u(z) as subharmonic functions in Q. For ~EM, u(~)=lim A(~, t, u)= -lim A(~, t, -u)= - [ -u(m. t~O t~O Hence u(z) is harmonic. Similarly, v(z) is harmonic and so f(z)=u(z) +iv(z)E(6'OO(Q). Hence 8f =0 in Q by continuity. 0 Remark. If M is an analytic subvariety, Corollary I.23 is the classical first Continuation Theorem of Riemann. Proposition 1.24. Let ljI(t) be an increasing convex function defined on [- 00, + 00) and let Proof Let XE't5~(Q), X;:;;O, let Proposition 1.25. Suppose Q is a domain in Proof Let n: z --+ xEIRn be the natural projection onto the real coordinates. Then cP extends in a natural way to a plurisubharmonic function on Q'=wxIRn, where w=n(Q). Let £>0 and let Q:={ZEQ': dQ,(z»£}. Then cp,(z)EPSH(Q:)n~CX)(Q:) and cp,(z) depends only on x. Furthermore, L(cp" w) n 82 = I 8 8CP, W k Wj' and if WEIR n, cP, is seen to be convex. Since a decreas• j,k=1 Xk Xj ing sequence of convex functions is convex, cP is convex, and since a convex function locally bounded from above is continuous, cP is continuous. 0 Corollaryl.26. Let Qc Proof Let Z=(ZI' ... , Zn)EQ. Then we can find a neighborhood W z of z such that we can define a branch logzk=vk+iv~ of logzk in W z for every k. For cpEPSH(Q), tjJ(vk)=t/i(vk+iv~)=cp(ev., ... , eVn ) is a plurisubharmonic function of the variable w=(v1 +iv;, ... , vn+iv~). By Proposition 1.25 it is a convex function of v=(v1 , ... , vJ Conversely, if tjJ(v) is defined in the open set w = {v: log rj < Vj < log rj'} and is a conve~ function of the variable v, we extend tjJ as a convex function on w + iIR n by tjJ (Vk + i v~) = tjJ (v k). By the remark following DefinitionI.2, t/iEPSH(w+iIRn). 0 Theoreml.27. Let QcIRm (resp. ii) {XEQ: cp(x) 1 Proof Clearly cpEL 1oc (Q), and by the Lebesgue Dominated Convergence Theorem, A(x, r, cp)= lim A(x, r, CPJ Thus A(x, r, cp) is a continuous function v~ 00 240 Appendix 1. Subharmonic and Plurisubharmonic Functions of X for 1'>0, A(x, 1', cp)ES(m Crespo A(x, 1', cp)EPSH(Q)] and is a convex and increasing function of 1', since this is true for A(x, 1', CPJ Hence !/l(x) = limA(x, 1', cp) is an upper semi-continuous function of x and !/JES(Q) Crespo 1"-0 !/JEPSH(Q)]. Moreover, for 1'>0 and all v, CPv(x)~cp(x)~A(x, 1', cp) so cp*(x)~A(x,r,cp) by the continuity of A(x,r,cp) and cp*(x)~!/J(x). Since lim A(x, 1', cp*)= cp* everywhere by upper semi-continuity, r~O !/J(x) = lim A(x, 1', cp) ~ lim A (x, 1', cp*)= cp* and !/J (x) = cp* (x). r--+ 0 J"-+ 0 It is a classical property of a function in Lloc(Q) that cp(x) = lim A(x, 1', cp) for almost all x, which proves (ii). r~ ° 0 Remark. From Theorem 1.27, we deduce. (1) Given a sequence CPv(X)ES(Q) Crespo CPvEPSH(Q)] locally bounded above, and !/J(x) = lim sup CPv(x) $ - 00, then !/J*(X)ES(Q) Crespo PSH(Q)] and v the set !/l(x) < !/J*(x) is of Lebesgue measure zero in Q. (2) The cones SeQ) and PSH(Q) are closed sets in LtcCQ) and given a Cauchy sequence CPvES(Q) Crespo in PSH(Q)] which converges to cpELloc(Q), !/l*(x)=[limsupcpv(x)]* is a limit of CPv in L\oc(Q) and CP=!/J=!/J* almost v-ct.) everywhere. To see 1), set CPn,p(x)=supcpv(x) for n~v~n+p; CPn,pES(Q) Crespo PSH (Q)]. By Theorem 1.27, if CPn = lim CPn. p~ CP:, the set p~co en= [x: CPn(x) =!/J(x) and if g=limcp~, !jt(x)~g(x) and the set [x: !/J(x) !/l (x) ~ lim sup A(x, r, cpvl = A(x, r, cp) ~ A(x, r, !/J) = A(x, r, !/J*) and !/J*(x)~A(x, 1', cp)~A(x, r, !jt*). Theorem 1.28 (Inverse Function Theorem for Plurisubharmonic Functions). Let QE ii) for ZED, we set b(z,m)={supr: 1'>0, M,p(z,r) Proof Part i) follows from Proposition 1.17. Thus, the points zED fall into two classes, those for which M aM aM We have -::;----'!'.=O, --,p>O. Since M and hence where so in this case, -log b(z, m) is plurisubharmonic. For the general case, we let CPv (z) be a sequence of ({jco plurisubharmonic functions which decrease to cp and let (\(z, In) be the associated functions on a domain Q' ~ D, for zoED'. Then -logbv(z, In) decreases to -logb(z, In), which is plurisubharmonic in D' by Proposition 1.3. D Appendix II. The Existence of Proximate Orders Theorem 11.1. Let M (r) be a continuous positive function for r > 0 such that . 10gM(r) I1m sup p < + 00. Then there exists a strong proximate order per) r-w logr such that M(r):£rP(r) for all r>O and M(rm)=r::.(r m ) for an increasing se• quence of values rm tending to + 00. . logq>(r) Proof Let q>(r)=M(r)·r- P so that hmsup O. We change variables HW logr by letting x = log rand y = log q> (r), so that . q>1(X) y=q>I(x)=logq>(expx) and hmsup--=O. X-HXJ x The idea of the proof is to construct piecewise a concave majorant to the curve y= q>1 (x) which coincides for a sequences of points Xm tending to infinity. This majorant will then be successively modified so as to have the differentiability properties required by the definition of a strong proximate order. The proof is divided into several steps. 1. First we construct a function lj; 1 (x) with the following properties: i) lj; 1 (x) is concave 11.. ) I·1m _._=lj;l(X) 0 , I·1m ,I,'1'1 ( x ) = +00 X--+OO x x-co iii) lim lj;~ (x) = 0 Let em be a sequence which decreases to zero. We choose by induction an increasing sequence of points Xm tending to + 00 and linear functions (Xm(x) of slope em such that (Xm(xm)=(Xm+l(Xm):£ -m and q>1(X» -(Xm(x)+m for x~xm: let (Xl(x)=-el x; we choose a point Xl such that elxl~1 and q>1(X 1)> -e1x 1 +1; having chosen xm and (Xm' we let (Xm+l(x)=(Xm(x)• em(x-xm); we choose xm+l so large that (Xm+l(xm+1):£ -(m+1) and q>1(X m + 1» -(Xm(xm+ 1)+m+1 and set t/fl(X) = -(Xm(x) for xm_l:£x:£xm · Then t/f I (x) satisfies (i), (ii), (iv), and (iii) except for the points Xm where t/f'l (x) does not exist. Thus, we modify t/f 1 in the following way: let 1m be the bisector of the obtuse angle formed by the lines y = (Xm(x) and y = (Xm+ I (x) Appendix II. The Existence of Proximate Orders 243 and let bm be the circle of radius om centered on 1m and tangent to (lm and (lm + 1; we use an arc on the circle for x between the x-coordinates of the two points of tangency, then for om sufficiently small, (i), (ii), and (iv) still hold; ljJ 1 (X)E~l (x) and (iii) also holds. 2. Suppose that ljJ(x) is a function which satisfies (i), (ii) and (iii) of 1. Then there exists a function Vex) such that (iv') lim Vex) = + 00 . Vex) . (v) hm -=0, hm O'(x)=O x_oo X X_CfJ 8" (x) (vi) lim -8'() =0 x_oo x (vii) 8(x) ~ ljJ(x) (viii) there exists an increasing sequence xm tending to + 00 such that xm is an extremal point for the curve y=ljJ(x), and furthermore 8(xm)=ljJ(xnJ Let sm be a sequence monotonically decreasing to zero. By induction, we shall find a sequence of points xm increasing to infinity and functions Om(x) defined on xm:;:;x:;:;xm +1 such that 8m (xm)=8m_ 1 (xm) and e~(xm)=8;n_l(Xm)' 0;:, (x) I - Ie~,(x) Let 81 (x) be a linear function with slope SI whose graph is tangent to the curve y=ljJ(x). The line y=a+s1 x lies above the curve for a large by (iii). If we decrease a in a continuous manner, we find an ao for which y = ao + SI x is tangent to the curve at [x~, ~f(X'I)] which is an extremal point by (iii). Let xo=O and Then 8m(x, elm)) approaches the function e~') + elm) (X - xm) asymptotically. Let ~m=O~_I(Xm)' which approaches elm-I) for large xm. We choose e~m) and e~m) (depending on the parameter elm)) so that 8m(xm)= 8m _ 1 (xm) and 8~,(xm) =8~_I(xm)' that is if Ym-l =Vm- 1 (x m- 1 ) • 1 1 e~n)=-(~m-elm)) and e~m)=ym_l +-(~m-elm)), Sm sm and we choose 0 < eim) < ~m so that e~m) > O. Then 8~(x) I I8~,(X) We choose Xm so large that ~m:;:;2eim-l) and Vm[x,~elm-l)J>IjJ(x) for x~xm' which is possible by (iii). Then there exists a e?n)<~eim-l) such that 244 Appendix II. The Existence of Proximate Orders the curve y=O",(x, e~n» meets y=lj;(x) tangentially at [lj;m(x~), x~]. Since y=Om(x, c\m» contains no line segments, [lj;m(x;n), x~J is an extremal point for y=ljJ(x). Furthermore, eyn)<2- mei1 ), so eim) goes to zero monotonically. Let 8m(x)=Om(x,cim» for x",~x~xm+l' Then 8(x) satisfies conditions (iv'Hviii) except for the points xm where 0" is not continuous. Since the points xm do not lie on the curve y=lj;(x), by changing 8(x) in a small neighborhood of xnp we can construct a new function O(x) whose second derivative lies between the upper and lower limits of 8" (x) at Xm and which still possesses the properties (iv'), (v) and (viii). Then I~: (x) I< 8 m _ 1 for xm ~ x ~ Xm+ l' so O(x) satisfies (vii) also. (x) Let 01 (x) be the function so constructed in (2) for lj; 1 (x). Let t/f 2 (x) be the smallest concave majorant of r~ oc lim p'(r)rlogr . I [O~(logr)-O~(logr)-(Oz(lOgr)-OI(lOgr»] 0 = hm r ogr = r~w r logr Furthermore, 1/ () O~ (log r) - Of; (log r) p r=--=------;o---=--- rZ log r + {[O~ (log r) - O~ (log r)] + [02(log r) - 01 (log r)J} 1'2 log r r2 (log 1')2 X{1+_2logr .}. It follows from (vii) and (vi) that 10;'(logr)I<10;(1ogr)l=o(1) and IO;(logr)1 = o(log r), i = 1, 2, so lim r210gr p" (1')= O. 0 r~ 00 Appendix III. Solution of the a-Equation with Growth Conditions The basic technique in the theory of functions of several complex variables is the solution of the a-equation, since a continuous function f defined in a domain Q c (Cn is hoi om orphic if and only if af = ° for the current af. We recall here the solution given by Hormander [BJ using LZ estimates and Hilbert space techniques for the equation au = g with ag = 0. We consider only (0,1) forms for g in this Appendix, which is sufficient for the problems treated in this book, and we refer the reader to [BJ for the general solution. 1. Basic Lemmas on Non-bounded Operators Between Hilbert Spaces Let H 1 and Hz be two complex Hilbert spaces with inner-products <, ) 1 and <, ) z respectively. We will consider an operator A from H 1 to Hz defined on a linear subspace D A of H l' called the domain of A, and a linear mapping A of DA into Hz. We would like to find the transpose of the operator A* with domain DA* in H 2 such that for XED A' YED A* (III,l ) Since A* Y will be uniquely determined only if DAis dense in H l' we shall always assume that this is the case. From (III,l) we then obtain I Thus, to the operator A with domain D A' we associate the subspace DA* defined to be the set of the elements YEH 2 for which there exists a constant Cy such that for every xEDA (III,2) For Y E DA*' we consider the mapping x -> A* Y from DA* into H 1 is linear. The operator A* with domain DA* is the transpose operator of A. 246 Appendix III. Solution of the a-Equation with Growth Conditions Proposition 111.1. The operator A* is closed, that is if Yn...... Yo, YnED A*' and zn=A*Yn ...... ZOEH1' then YoEDA* and zo=A*yo. Proof Since zn is a Cauchy sequence in HI' there exists an M such that Ilz,,11 ;£M, and hence I We would like to define (A*)* and verify that (A*)*=A. To do so, we introduce the additional hypothesis that A is closed and show that DA* is dense in H 2. Then (A*)* can be defined, and (A*)* = A (which implies that D A= D(A*)* when A is closed). We introduce the product space H = HI X H 2 and fI = H 2 X HI and the mappings Bl and B2 of H into fI and fI into H respectively given by B 1 (x,y)=(y, -x) and B2(y,x)=(x, -y). The graphGA of A is the set (x, Ax) in H with xEDA and the graphGA* of A* is the set (y, A* y) in fI with YED A*. We equip H with the inner product «x,y),(u,v)= Proposition 111.2. We have (III,3) Proof The relation Proposition 111.3. If GA is closed, then A* is dense in H 2' DA = D (A*)*' and A = (A*)*. Proof From the closure of GA, we see that Bl (GA) =(B1(GA)).L.L, and from Proposition II1.2, we see that (B 1(G A)).L=(GA*).L. Hence GA = B 2 (B 1(G A)) =B2 (Gi*) =(B2(G A*).L) and (III,4) GA = (B 2 (GA*)).L in H. Let uEH2n(DA*).L. Then «O,u), (A*y,-y)=O in H for all YEDA*, and so (0, u)EB2(GA*).L. Thus, by (III,4), (0, U)EGA and u=A(O)=O, from which it follows that A* is dense in H 2. By applying (III,3) to A*, we obtain G(A*)*=(B 2 (GA*)).L. Thus by (I1I,4) , GA=G(A*)*' from which it follows that DA=D(A*)* and A = (A*)*. D Appendix III. Solution of the a-Equation with Growth Conditions 247 Lemma 111.4. Let A be a closed operator from a dense subspace DA of a Hilbert space H j into a closed subspace F of a Hilbert space Hz. Then F=A(DA) if and only if there exists a constant C~O such that for every YEF nD1, IIyl12 ~ ClIA*ylll' Proof Let zEF. The solution of the equation Ax=z is equivalent to the existence of an x such that Suppose that F = A(DA)' Let B = {y: YEF n DA*' IIA* yl1 1~ I}. We shall show that B is a bounded set in H 2' The space A (D A) is closed and therefore is a Hilbert space; then for YEB and zEF=A(DA) we obtain: Thus the family B is pointwise bounded on F. By the Banach-Steinhaus Theorem it is equicontinuous and uniformly bounded on the unit ball in F by a constant C. This implies that \(y, II~II)\ ~ Cor IIyl12 ~ C. D Lemma 111.5. Let A be a closed operator defined on a domain DA dense in HI and let F be a closed subspace of the Hilbert space H 2 such that A (D A) c F. Suppose that there exists C>O such that Ilyllz~ CIIA*ylll for every 1 YEFnDA*. Then for vEH1 n[A- (0)l\ there exists wEDA* such that A*w=v and IlwI12~CllvII1' Proof Since A is closed, if xnEDA' xn~xO' and Axn=O, then xoEDA and Axo=O, so A-I(O) is a closed subspace of HI' Now Ax=O is equivalent to 2. Inequalities for the t3-equation Let cP be a continuous function on Q c (C" and let LZ (cp) be the completion of '6'~(Q) for the norm induced by the inner product ITCfW;£ C T S Iflze-(P(Z)dr(z) for fE'6'~(Q). Q By L~o.l)(cp) (resp. ~O,Z)(cp)) we will mean the space of (0,1) forms " f = L J;dzi (resp. the space of (0,2)-forms f = L .t;jdzi /\ dz) such that i= 1 i Proposition 111.6. The operator A = 3 is a closed densely defined operator. Proof Since '6'~ (Q) is dense in L2 (cp 1)' A is densely defined. If f" -> fo in LZ(CP1)' then f" -> fo in Lioc(Q), Since derivation is a continuous operation in the space of distributions, Af" -> Aj~ as a current. Thus, if Aj~ -> go III L70, 1)(CPZ)' Afo = go for the currents, and A is closed. 0 We shall also consider the operator B = 3 which maps the space L70,1)(CPz) into L7o,z)(CP3) given by f'd-) '" (8h 8·t;)d- Jj- 8-(~.L.. Ji Zi =.L... ~-8-. Zi/\UZj' ,=1 ,<] OZ, Z] For what follows, we choose a function a(z)E'6'~ (B o) defined in the unit ball Bo of (Cn and such that S a(z)dr(z) = 1, and we set a£(z)=CZna G')' Lemma 111.7. Given gELZ(Q) with compact support, then g£ = g*a£ is in '6'~ and lim Ilg£-gllu=O. £~ 0 Appendix III. Solution of the a-Equation with Growth Conditions 249 Proof Since g,(z) = S g(z + z')IX,(z')dT(Z') and g has compact support, we may differentiate under the integral, which shows that g, is in 't'g'. If u is continuous, it follows from the formula ue(z) - u(z) = J(u(z - 32') - u(z)) IX,Cz')dT (z') and the uniform continuity of U that U e converges uniformly to u. Since S 1X,(z)dT(Z) = 1, it follows from Minkowski's Inequality that in L2 norms Iluell;£ Iluli. For any 11>0, we can find a continuous function v with compact support such that Ilu -vii < 11. Thus, Ilu, -v,1I < 11 and lim sup Ilu, -ull ;£lim sup Ilu, -v, II + Ilu -vii + lim sup Ilv, -vii ;£ 211, £-0 &-0 E-O since v, -+ v uniformly. o II Now we shall calculate explicitly A*. Let gE't'g'(Q) and f = L ;;dzi in ~O.l)(Q). Then if fEDA*' j~l J(A*f)ge-'PldT = (III,6) Proposition 111.8. Let Ko be a fixed compact set and 13m a sequence of functions in 't';(Q) such that O;£f3m;£ 1, 13m = 1 for zEKo, and such that for every compact subset K, there exists mK such that m~mK implies 13m =1 on K. Suppose that Cf12E't'I(Q) and n (III,7) e-'Pj+ 1 L lof3m/ozkl2 ;£e-'Pj, j= 1, 2. k~ I Then the (0, 1) forms with coefficients in 't'g' (Q) are dense in DA* n DB for the norm IllfIIl= IIA*flll + Ilf112+ IIBfI13' where II III is the norm in V(Cf1!), II 112 the 110rm in L~O,I)(Cf12)' and II 113 the norm in L~O,2)(Cf13)' Proof Since 13m has compact support for all 111 and is in 't'w, 813m /\ f and f3m8f have coefficients in L7o.2/Cf13) if fEjJB' Furthermore, B(f3mf)-f3m(Bf)=of3m/\f, and from (III,7), we have IB(f3mf) - f3m(BfWe-(P3;£ Ifl2 e-(PZ. Since B(f3mf) - f3m(Bf) converges to zero almost everywhere, it follows from the Lebesgue Dominated Convergence Theorem that m---"Cfj Suppose that fEDA * and gEDA" Then and 1<1, A(f3mg)21 = I It then follows from (III,7) and Schwarz' Inequality that IA*(f3nJ) - f3",{A* fW ~L 1.0 2e( Thus the elements of DBnDA* with compact support are dense for the norm 111111· Let fEDA*nDB have compact support. Then f*rxc for 6<1 has its support contained in a fixed compact set, and since (8 + a) U*rxc) = [(8 + a) f]* rxc + aU *rx.l-(af)*rxe , As above, the right hand side converges to (8+a)f +af -af in L2(1) and hence lim IIA*U*rxJ-A*fll l =0, 0 c~O Theorem 111.9. Let Ko be a fixed compact subset of Q and f3mE((j~J(Q) such that O~f3m~l, f3m=l for zEKo and such that for every compact subset KeQ, there exists mK such that for m~mK' f3m=l on K. Let tj;EC(j2 such that z z m laf3 l n 3 (z) L 3 m ~eljJ. Let cpEPSH(Q)nC(j2 such that L CP,'7 Wj\;j\~ C(z) Ilwllz k=l Zk j,k=l 3zjc~k for a function C and for all WE~n. Set CPl =cp-2tj;, CP2=CP-tj;, ¢3=CP. Then S (C - 2Iatj;llfI2)e- (III, 8) From (III,6), we obtain since CPt -CP2 = -lj;. Using the inequality for vectors Ila-bI12~21IaI12+21IbI12, we see that n _ n S I (5jfj(5Jke-q>dr~21IA*fllt+2S I Ifl·18lj;1 2 e-q>dr. j.k~1 j~1 An easy calculation shows that = I \o.~ \2 _ I o~ ~~ . .. 1 1 1,J= oz.J l,j=.. oz.J oz·I Adding together these two results, we obtain Suppose now that the coefficients of f are in 'tJ';". An integration by parts then gives and Another integration by parts yields -Sf.!; (Ofk) -'I'd =SOfj.Ofk -'I'd ,Uk 0- e r ~ 0 e r, z; OZk Zj 252 Appendix TIL Solution of the a-Equation with Growth Conditions and so . _ ~ _'I' n jaf j j2_ Definition 111.10. A domain QE Lemma 111.11. Let Q be a pseudoconvex domain in for every zEQ, WE (II1,9) provided ~ is finite. Proof Let y(z) be the function of Definition 1lI.10 associated with Q. For any fixed r, we can assume that the functions 13m of Proposition III.S are such that 13m == 1 on Qr+ 1 for all m. We can then find l/t ~ 0, 00 11 jaf3jZ I I a- m ~eljJ and such that l/t=0 in Qr+l' ' m= 1 k= 1 Zk Let x(r) be an increasing convex function such that X(y(z)) ~ 2l/t(z) and n a2y x'(Y(z)) j,2l azjazk wjwk~2Ial/tlzllwI12. Let cp~ =CP+X(Y)-2l/t, CP;=CP+X(Y)-l/t, and cP~=cP+X(y). By Theorem IIL9, for fEDA*nD B with coefficients in 'fj;{', S C(z)lf(zWe- Tg = 1/2. Then by the Cauchy-Schwarz Inequality, we have I Let f = fl + f2' where Bfl =0 and fl is orthogonal in ~o, I)(CP'~) to the kernel of B, Since the range of A is contained in the kernel of B, fl is orthogonal to the range of A, so A*fl=O, Since ag=o, Theorem 111.12. Let Q be a pseudoconvex domain in (Cn and let cpEPSH(Q). for every gEL~o.I)(CP) such that ag=o, there exists a function u such that au=g and (III,10) S lu1 2 (1 + Ilzlll)-le-"'dr~ S IgI 2e-(Pdr. Q Q Proof Let y(z) be the function of Definition lILlO. Let Qj={ZEQ:Y(Z) By Proposition Ll5, we can find cpjEPSH(Qj)n(~"Xl(Q) such that CPj decreases to cP as j tends to infinity. Let l/Ij = cP j + 2log (1 + liz 11 2). Then, since ± a2lO~(1 ~ Ilz112) Wj~vk=(1 + IlzI12)-2(lIwI12(1 + IlzI1 2)-«w, z>?) j,k~ 1 Zja"'k ~(1 + IlzI12)-2IiwI12 we can take C(z)=2(1 + IlzI12)-2 in Lemma nLll. Thus, for every j, we can find uj defined in Qj such that S lul(1+ Ilzlll)-2e-"'jdr~ S IgI2e-"'jdr~S Iglle-"'dr. ~ ~ Q Since the sequence cP j is uniformly bounded above on every compact subset of Q, we can find a subsequence ujk such_ that ujk converges weakly to a function U in L2(Q[) for every l. Then au=g, since differentiation of a distribution is continuous in L\oc and S lule-"'jdr~ S IgI 2e-(Pdr Q, Q for all j and I, so S luI 2e-(P(1 + IlzI12)-2dr~ S IlgI1 2r"'dr. o Q Q Bibliography A. Gunning, R., Rossi, H., Analytic Functions of Several Complex Variables, Prentice-Hall, Englewood Cliffs, N.J. (1965) B. Hiirmander, L., An Introduction to Complex Analysis ill Several Variables, North-Holland, Amsterdam (1973) C. Lelong, P., F onctions Plurisousharmoniques et Formes Dij}erentielles Positives, Dunod, Paris (1968) and Gordon and Breach, New York (1969) D. Levin, B.Ja., Distribution of Zeros of Entire Functions, Trans. Math. Monographs, Vol. 5, American Math. Soc., Providence, R.1. (1964) E. de Rahm, G., Varietes difJerentiables, Hermann, Paris (1960) F. Schwartz, L., l1u'orie des Distributions, Hermann, Paris (1966) G. Treves, F., Lincar Partial Differential Equations with Constant Coefficients, Gordon and Breach, New York (1966) H. Wells, R.O., Dij}erelltial Analysis on Complex Manifolds, Springer-Verlag, New York (1980) Agarwal, A.K. 1. On the properties of an entire function of two complex variables, Canad. J. Math. 20 (1968), 51-57 2. On the geometric means of entire functions of several complex variables, Trans. Amer. Math. Soc. 151 (1970),651-657 Agranovic, P.Z. 1. The existence of a function hoi om orphic in a cone with a prescribed indicator and having proximate order (Russian), Teor. Funkcii Funkcional Anal. i. Prilozen Vyp. 24 (1975), 3-15 2. Functions of several variables with completely regular growth (Russian), Teor, Funkcii Funkcional Anal. i. Prilozen Vyp. 30 (1978), 3-13 Agranovic, P.Z., Ronkin, L.1. 1. Functions of completely regular growth of several variables, Ann. Polon. Math. 39 (1981), 239-254 Aizenberg, LA. 1. The general form of a linear continuous functional in spaces of functions holomorphic in convex domains in 2. Fonctions plurisousharmoniques differences de deux fonctions plurisousharmoniques de type exponentiel, C.R. Acad. Sci. Paris 252 (1961), 499-500 3. Fonctions entieres de p variables et fonctions plurisousharmoniques it croissance tres lente. J. Analyse Math. Jerusalem 9 (1971-72), 347-361 4. Ouverts d'exclusion dans ([P (p S 2) pour les fonctions entieres it croissance lente, C.R. Acad. Sci. Paris, Ser. A-B, 274 (1972),1915-1918 5. Quelques applications de la methode des "boules d'exclusion" dans ([P (Armenian and Russian Summaries), Izv. Akad. Nauk. Armjan SSR, Ser. Math. 8 (1973), N° 4,346,306-320 Avanissian, V., Gay, R. 1. Sur les fonctions entieres de plusieurs variables, C. R. Acad. Sci. Paris, Ser. A-B, 266 (1968), 1187-1190 2. Sur une transformation des fonctionnelles analytiques et ses applications aux fonctions entiere de plusieurs variables, Bull. Soc. Math. France, 103 (1975), N° 3,341-384 Ba vrin, 1.1. 1. The nature of a pair of analytic functions, one of which is entire, which are univalent in the space of two complex variables (Russian), Moskov. Oblast. Pedagog. Inst. Uc. Zap. 57 (1957), 33-37 Berenstein, C.A. 1. The number of zeros of an analytic function in a cone, Bull. Amer. Math. Soc. 81 (1975), 213-214 Bercnstein, C.A., Dostal, M.A. 1. A lower estimate for exponential sums, Bull. Amer. Math. Soc. 80 (1974) 687-691 2. The Ritt Theorem in several variables, Ark. Mat. 12 (1974), 267-380 Berenstein, c.A., Taylor, B.A. 1. Interpolation problems in ([n with application to harmonic analysis, J. Analyse Math. Jerusalem 38 (1980), 188-254 Berndtsson, B. 1. Zeros of analytic functions of several variables and related topics, Thesis, Univ. of Goteborg, 1977 2. Zeros of analytic functions of several variables, Arkiv for Mat. 16 (1978) 3. A note on Pavlov-Korevaar interpolation, Nederl. Akad. Wetensch. Proc. ser. A. 81 (1978) Bernstein, S.N. 1. On entire functions of finite degree of several complex variables (Russian), Doklady Akad. Nauk. SSR (N.S) 60 (1948), 949-952 Bieberbach, L. 1. Beispiel zweier ganzer Funktionen zweier komplexer Variablen, welche eine schlicht volu• mentreue Abbildung des R4 auf einen Teil seiner selbst vermitteln, Preuss. Akad. Wiss. Sitzungsbcr. (1933),476-479 Bitlyan, I.F., Gol'dberg, A.A. 1. The Wiman-Val iron Theorems for integral functions of several complex variables (Russian, English summary). Vestnik Leningrad Univ. 14 (1959), Vol. 13, 27-41 Boas, R.P. 1. Entire Functions, Academic Press (1954), New York Bochner, S. 1. Entire functions in several variables with constant absolute values on a circular uniqueness set. Proc. Amer. Soc. 13 (1942), 117-120 Bombieri, E. 1. Algebraic values of meromorphic maps, Invent. Math. 10 (1970), 248-263 Bombieri, E., Lang, S. 1. Analytic subgroups of group varieties, Invent. Math. 11 (1970), 1-14 Borel, E. 1. Le90ns sur les series it termes positifs, Gauthier-Villars, Paris 1902 Bose, S.K., Kumar, K. 1. On a class of Dirichlet series over ([2 Ann. Soc. Sci. Bruxelles, Ser. I, 89 (1975), N° 4, 509- 521 256 Bibliography Bose, S.K., Sharma, D. 1. Integral functions of two complex variables, Compositio Math. 15 (1963), 210-226 Carlson, J. 1. Some degeneracy theorems for entire functions with values in an algebraic variety. Trans. Amer. Math. Soc. 168 (1972), 273-301 2. A remark on the transcendental Bezout problem, Value Distribution Theory (Part A), Proc. Tulane Univ. Program on Value Distribution Theory in Complex Analysis, Marcel-Dekker, New York (1974), 133-143 3. A moving lemma for the transcendental Bezout problem, Ann. of Math. 103 (1976), 305-330 4. A resnlt on value distribution of hoI om orphic maps of (Cn --> (Cn, Several complex variables (Proc. Sympos. Pure Math., Vol. XXX, Part 2, Williams ColI. Williamstown, Mass. 1975), Amer. Math. Soc., Providence, R.l. (1977), 225-227 Carlson, J., Griffiths, P.A 1. The order functions for entire holomorphic mappings, Value Distribution Theory, (Part A) Proc. Tulane Univ. Program on Value Distribution Theory in Complex Analysis, Marcel• Dekker, New York (1974), 225-248 Chern, S.S. 1. The integrated form of the first main theorem for complex analytic mappings in several complex variables, Ann. of Math. 2, 71 (1960), 536-551 Chon, C.c. 1. Sur Ie module minimal des fonctions entieres de plusieurs variables complexes d'ordre inferieur it 1, C.R. Acad. Sci. Paris, Ser. A-B, 267 (1968), 779-780 Cornalba, M., Shiffman, B. 1. A counterexample to the "Transcendental Bezout Problem", Ann. of Math., Vol. 96 (1972), 402-406 Dalal, S.S. 1. On the order and type of integral functions of several complex variables, J. Indian Math. Soc. (N.S.) 33 (1969), 215-220 Demailly, J.P. 1. Formules de Jensen en plusieurs variables et applications arithmetiques, Bull. Soc. Math. France 110 (1982), 85-102 2. Sur les nombres de Lelong associes it I'image directe d'un courant positif ferme, Ann. Inst. Fourier Grenoble 32, N° 2 (1982), 37-66 Deny, J., Le1ong, P. 1. Sur une generalisation de l'indicatrice de Phragmen-Lindelof, C.R. Acad. Sci. Paris 224 (1947), 1046-1048 2. Etude des fonctions sousharmoniques dans un cylindre 0\1 dans un cone, Bull. Soc. Math. France (1947), 89-112 Dikshit, G.P., Agarwal, AK. 1. On the means of entire fnnctions of several complex variables. Ganita 21 (1970), N° 1, 75-85 Dzafarov, AS. 1. Some inequalities for entire function of finite degree (Russian) Izv. Vyss. Ucebn. Zaved Matematika (1960), N° 1,14,103-115 2. Some generalizations of Berenstein's inequality for entire functions of finite degree (Azer• baijani), Akad. Nauk. Azerbaidzan. S.S.R. Trudy Inst. Math. Meh. 1 (9) (1961), 87-98 3. Inequalities between various weighted norms for entire functions of exponential type (Rus• sian), Izv. Akad. Nauk. Azerbaidzan S.S.R. Ser. Fiz. Mat. Techn. Nauk. (1963), N° 2, 17-25, MR 28-244 4. A generalization of inequalities of Ehrenpreis, Malgrange, Hormander, and Rosenbloom on entire functions of exponential type (Russian), Akad. Nauk Azerbaidzan SSR, Dokl. 19 (1963), N° 5, 3-6 5. Generalization of an inequality of R. Boas for entire functions of exponential type (Russian), Azerbaidzan Gos. Univ. Ucen Zap. Ser. Fiz. Mat. i Him Nauk (1964), N° 4, 3-9 6. Inequalities with a weight for entire functions of finite order (Russian), Akad. Nauk Azer• baidzan, SSR Dokl. 20 (1964), N" 12, 3-6 Bibliography 257 7. Inequalities for entire functions belonging to a certain class (Russian), Studia Sci. Math. Hungar I (1964), 17-25 Dzafarov, AS., Ibragimov, 1.1. 1. Some inequalities with weight for entire functions of finite degree (Russian), Uspehi, Mat. Nauk. 19 (1964) N° 6,120, 147-154 Dhbajyan, M.M. 1. On the theory of some classes of entire functions of several variables (Russian), Akad. Nauk Armyan. SSR, Izv. Fiz. Mat., Estest. Tchn. Nauk 8 (1955), N° 4,1-23 2. On integral representation and expansion in generalized Taylor series of entire functions of several complex variables (Russian), Mat. Sb. N.S. 41, 83 (1957), 257-276 Ehrenpreis, L. 1. A fundamental principle for systems of linear differential equations with constant coefficients and some applications, Proc. Internat. Sympos. Linear Spaces (Jerusalem 1960), Jerusalem Ac. Press, Pergamon Press, Oxford (1962),161-174 Eremine, S.A 1. Sur des fonctions entieres de deux variables (Russian, French), Ukrain Mat. Z. 9 (1957), 30-43 Evgrafov, M.A 1. Integral representation of functions of exponential growth (Russian). Dok!. Akad. Nauk SSSR 168 (1966), 512-515 Fatou, P. 1. Sur certaines fonctions complexes de deux variables. C.R. Acad. Sci. Paris 175 (1922), 1030- 1033 2. Sur les fonctions meromorphes de deux variables. C.R. Acad. Sci. Paris 175 (1922), 862-865 Favarov, SJa. 1. The addition of the indicator of entire and subharmonic functions of several variables (Russian), Mat. Sb. (N.S.) 105, 147 (1978), N° 1, 128-140 Filmonova, L.A 1. A certain condition for the representability of an entire function of two complex variables by a double Newton series (Russian), Ural. Gos. Univ. Mat. Zap. 8, tetrad' 4, 100-108, 136 (1974) Fuks, B.A 1. Introduction to the theory of analytic functions of several complex variables, Translation Math. Monographs 8, Amer. Math. Soc., Providence, R.I., 1963 2. Special chapters in the Theory of analytic functions of several complex variables, Translation Math. Monographs 14, Amer. Math. Soc., Providence R.I., 1965 Gavrilova, R.M. 1. The representation of entire functions of two complex variables by Dirichlet series (Russian), Teor. Funkcii Funkcional. Anal. i Prilozen. Vyp. 10 (1970).71-78 Gece, F.1. 1. Systems of entire functions of several variables and their applications to the theory of differential equations (Russian), Izv. Akad. Nauk. Armjan SSR, Ser. Fiz. Mat. Nauk 17 (1964), N°2, 17-46 2. Growth characteristics of entire functions of several complex variables (Russian), Dok!. Akad. Nauk SSSR, 164 (1965), 487-490 (English translation: Soviet Math. Dok!. 6 (1965), 1242-1246 3. On a certain class of entire functions of several variables (Russian), ·Ukrain Mat. Z. 18 (1966), N° 3, 13-27 4. Refined growth characteristics of entire functions of several complex variables (Russian, Lithuanian, and German summaries), Litovsk Mat. Sb. 8 (1968), 461-488 5. An investigation of the growth of entire and holomorphic functions of several complex variables by means of directed characteristics (Ukranian; English and Russian summaries), Dopovidi Akad. Nauk Ukrain SR, Ser. A (1975), 105-110 Gece, F.l., Kurei, AI. 1. The entire solutions of linear partial differential equations of infinite order (Russian; Ar• menian and English summaries), Izv. Akad. Nauk. Armjan. SSR, Ser. Mat. 8 (1973) N° 2, 123-143 258 Bibliography Gol'dberg, A.A. I. Elementary remarks on the formulas defining order and type of functions of several variables (Russian), Akad. Nauk. Armjan SSR, Dokl. 29 (1959),145-151 Gopola, KJ., Nagaraja Rao, l.H. I. On orders and types of an entire function, J. Austral. Math. Soc. 15 (1973), 393-408 Griffiths, P.A. I. On the Bezout problem for entire analytic sets, Ann. of Math. 100 (1974), 533-552 Griffiths, P.A., King, J. I. Nevanlinna theory and holomorphic mappings between algebraic varieties, Acta Math. 130 (1973),145-220 Gromov, v.P. I. The representation of functions by double Dirichlet sequences (Russian), Mat. Zametki 7 (1970),53-61; English translation. Math. Notes 7, 1970,33-37 Gross, F. I. Entire functions all of whose derivatives are integral at the origin. Duke Math. J. 31 (1964), 617-622 2. Generalized Taylor series and orders and types of entire functions of several complex variables. Trans. Amer. Math. Ser. 120 (1945), 124-144 3. Entire functions of several variables with algebraic derivatives at certain algebraic points. Pacific J. Math. 31 (1969),693-701 Gruman, L. I. Entire functions of several variables and their asymptotic growth. Ark. Mat. 9 (1971), 141- 163 2. The regularity of growth of entire functions whose zeros arc hyperplanes. Ark. Mat. 10 (1972),23 -31 3. The growth of entire solutions of differential equations of finite and infinite order. Ann. Inst. Fourier (Grenoble), 22 (1972), N" 1,211-238 4. Some precisions on the Fourier-Borel transform and infinite order differential equations. Glasgow Math.J. 14 (1973),161-167 5. Infinite order differential equations in Banach spaces of entire functions. J. London Math. Soc., 2 (1974), 492-500 6. Interpolation in families of entire functions in Hahn, K.T. I. A remark on integral functions of several complex variables. Pacific J. Math. 26 (1968), 509- 513 Hantler, S.L. 1. Estimates for the a-Neumann operator in weighted Hilbert spaces. Trans. Amer. Math. Soc. 217 (1976), 395-406 Hengartner, W. 1. Proprietes des restrictions d'une fonction plurisousharmonique ou entiere dans Kamthan, P.K., Jain, P.K. 1. Remarks on the geometric means of entire functions of several complex variables, Riv. Mat. Univ. Parma (3) (1972), 113-117 Kardas, AI., Culik, I.I. 1. Majorant properties and Newton diagrams of entire functions of two complex variables (Ukranian; English and Russian summaries), Dopovidi Akad. Nauk. Ukrain. RSR, Ser. A (1969),583-586,665 Kioustelidis, J. 1. Eine einheitliche Methode zur HerIeitung von Reihenentwicklungen fUr ganze Funktionen vom Exponential-typ, Composito Math. 26 (1973), 203-232 Kiselman, C.O. 1. On unique supports of analytic functionals, Ark. Math. 6 (1966), 307-318 2. On entire functions of exponential type and indicators of analytic functionals, Acta Math. 117 (1967),1-35 3. The growth of restrictions of plurisubharmonic functions, Mathematical analysis and appli• cations, Part. B, Adv. in Math. Supp!. Stud. 76, Academic Press, New York (1981), 438-454 Kneser, H. 1. Zur Theorie der gebrochenen Funktionen mehrerer Vedinderlichen, J ahresbericht der Deutschen Math. Vereinignng, t. 48 (1948), 1-28 Kobeleva, N.L. 1. The relation between the growth of an entire fnnction of two complex variables and the distribution of singularities of its associated functions (Russian), Izv. Vyss. Zaved Mate• matika (1962), N° 3, (28), 59-66 Korevaar, J., Hellerstein, S. 1. Discrete sets of uniqneness for bounded holomorphic functions f (z, w). Entire functions and Related Parts of Analysis (Proc. Sympos. Pure Math., La Jolla, Calif., 1966), 273-284. Amer. Math. Soc., Providence, R.I. (1968) Korobelnik, J.F., MOrZakov, V.V. 1. A general form of the isomorphisms that commute with a differentiation operator in spaces of entire functions of slow growth (Russian), Mat. Sb. (N.S.), 91, (133) (1973), 475-487, 629. English translation: Math. Notes 20 (1973), 493-505 Kozmanova, AA ]. Polya's theorem for integral functions of two complex variables (Russian), Dokl. Akad. Nauk. SSSR (N.S.) 113 (1957),1203-1205 Kramer, R.A 1. Zeros of entire functions in several complex variables. Trans. Amer. Math. Soc. 176 (1973), 253-261 Kravcenko, F.G. 1. Analytic functions of roots of polynomials (Russian; English summary), Vyeis!. Prik!. Mat. (Kiev), typo 7 (1969), 77-93 Kujula, R.O. 1. Functions of finite ie-type in several complex variables, Bull. Amer. Math. Soc. 75 (1969), 104-107 2. Functions of finite X-type in several complex variables, Trans. Amer. Math:Soc. 161 (1971), 327-358 Kurita, M. 1. A theorem on the value-distribution of a complex analytic mapping (Japanese). Sugahu 16 (1945), ] 95-202 Lal, J., Dikshit, G.P. 1. The Phragmen-Lindelof principle for functions of several complex variables, Riv. Mat. Univ. Parma, (2), 6 (1945), 283-286 Lang, S. 1. Introduction to transcendental numbers, Addison-Wellesley (1966) Lelong, P. 1. Sur l'ordre d'une fonclion entiere de deux variables, C.R. Acad. Sci. Paris 210 (1940), 470 Bibliography 261 2. Sur quelques problemes de la theorie des fonctions de deux variables complexes. Ann. Ec. Norm. 58 (1941), 83-177 3. Sur les valeurs lacunaires d'une relation Ii deux variables complexes, Bull. Sci. Math. 56 (1942),103-112 4. Sur la capacite de certains ensembles de valeurs exceptionnelles. C.R. Acad. Sci. Paris t. 214 (1942),992 5. Definition des fonctions plurisousharmoniques, C.R. Ac. Sci., t. 215, 398-400 (1942) 6. Sur les suites de fonctions plurisousharmoniques, C.R. Ac. Sci., t. 215, 454-456 (1942) 7. Les fonctions plurisousharmoniques, Ann. Ec. Norm., t. 62, 301-338 (1950) 8. Proprietes metriques des varietes analytiques complexes definies par une equation, Ann. Ec. Norm., t. 67, 393-419 (1950) 9a. Sur la representation d'une fonction plurisousharmonique Ii partir d'un potentiel, C.R. Ac. Sci., t. 237, 691-693 (1953) 9b. Sur l'extension aux fonctions entieres de n variables, d'ordre fini, d'un developpement canonique de Weierstrass, C.R. Ac. Sci., t. 237, 865-867 (1953) 9c. Sur l'etude des noyaux primaires et un theoreme de divisibilite des fonctions entieres de /1 variables, C.R. Ac. Sci., t. 237, 1379-1381 (1953) 10. Integration of a differential form on an analytic complex subvariety, Proc. Nat. Ac. of Sciences, 43, 246-248 (1957) 11. Fonctions entieres (/1 variables) et fonctions plurisouharmoniques d'ordre fini dans Lozinski, S. 1. A generalization of a theorem of S. Bernstein (Russian), Doklady Akad. Nauk SSSR (N.S.), 55 (1947), 9-12 Lunc, G.L. 1. The convergence of certain general series in the space of several complex variables (Russian), Sibirsk. Mat. Z. 13 (1972), 467-472 Maergoiz, L.S. 1. A property of the indicator of an entire function of several variables (Russian), Izv. Vyss. Uiebn Zaved Matematika (1964), N" 6 (43), 104-115 2. On the question of the relation between various definitions of orders of entire functions of several variables (Russian), Sibirsk. Mat. Z. 7 (1966), 1268-1292 3. Some properties of convex sets and their applications to the theory of the growth of convex and entire functions (Russian), Sibirsk. Mat. Z. 9 (1968), 577-591 4. Scales of growth of entire functions of several variables (Russian), Dok!. Akad. Nauk. SSSR 192 (1970), 495-498. English translation: Soviet Math. Dok!. 11 1970),662-666 5. A fnnction of the orders and scale of growth 6f entire functions of several variables (Russian), Sibirsk. Mat. Z. 13 (1972), 118-132. English translation: Siberian Math. J. 13 (1972), 83-93 (1973) 6. Types and their associated order of growth for entire functions of several variables (Russian), Ook!. Akad. Nauk. SSSR, 213 (1973), 1025-1028. English translation: Soviet Math. Dok!. 14 (1973), 1846-1850 (1974) 7. Functions having the types of an entire function of several variables with regard to its directions of growth (Russian), Sibirsk. Math. Z. 14 (1973), 1037-1056, 1157. English trans• lation: Siberian Math. J. 14 (1973) 723-736 (1974) 8. The multidimensional analogue of the type of an entire function (Russian), Uspehi Mat. Nauk. 30 (1975), N" 5, (185), 215-216 Maergoiz, L.S., Yakolev, E.!. 1. Growth of convex and entire functions of infinite order with respect to the totality of variables, Some problems of multidimensional complex analysis (Russian), Akad. Nauk. SSSR, Sibirsk Otdel, Inst. Fiz. Krasnagarsk (1980), 79-83 Magnus, A. 1. On polynomial solutions of a differential equation, Math. Scand. 3 (1955), 255-260 Malgrange, B. 1. Existence et approximation des solutions des equations aux derivees partielles et equation de convolution, Ann. Inst. Fourier 6 (1956), 271-355 Mamedhanov, OJ. 1. Some properties of an entire function of finite degree in a generalized Lebesgue space (Russian), Functional Anal. Certain Problems Theory of Difference. Equations and Theory of Functions (Russian), 150-160, Izdat. Akad. Nauk. Azerbaidzan SSR, Baku (1967) 2. Inequalities for positive entire functions in a generalized Lebesgue space (Russian), Dok!. Akad. Nauk. SSSR, 157 (1944), 526-528 Martineau, A. 1. Indicatrices des fonctions analytiques et inversion de la transformation de Fourier-Borel par la transformation de Laplace, S.c. Acad. Sci. Paris, 255 (1962), 2888-2890 2. Indicatrices des fonctionnelles analytiques et inversion de la transformee de Fourier-Borel par la transformation de Laplace, C.R. Acad. Sci. Paris, 255 (1962), 1845-1847 3. Sur les fonctionnelles analytiques et la transformation de Fourier-Borel, Jour. Analyse Math (Jerusalem) XI (1963),1-164 4. Indicatrices des croissances des fonctions entieres de N-variables, Invent. Math. 2 (1966), 81- 86 5. Indicatrices de croissance des fonctions entieres de N-variables; Corrections et complements, Invent. Math. 3 (1967), 16-19 6. Unicite du support d'une fonctionnelle analytique: un theoreme de C.O. Kiselman, Bull. Sci. Math. (2),91 (1967), 131-141 7. Fonctionnelles analytiques non-lineaires et representation de Polya pour une fonction entiere Bibliography 263 de n-variables de type exponentiel, Seminaire P. Lelong (Analyse), Annee 1970, 129-165, Lecture Notes in Math., Vol. 205, Springer (1971) 8. Equations differentielles d'ordre infini, Soc. Math. France 95 (1967), 109-154 Meteger,1. 1. Local ideals in a topological algebra of entire functions characterized by non-radial rate of growth, Pacific J. Math., 51 (1974),251-256 Maude, R. 1. Exceptional sets with respect to order of integral functions of two variables, Proc. Cam• bridge Philos. Soc. 53 (1957), 323-342 Molzon, R.E. 1. Capacity and equidistribution for holomorphic maps from ([:2 to ([:2, Proc. Amer. Math. Soc. 71 (1978), N° 1, 46-48 2. Sets omitted by equidimensional holomorphic mappings, Amer. J. Math. 101 (1979), N° 6, 1271-1283 3. The Bezout problem for a special class of functions, Michigan Math. J. 26 (1979), NU 1,71-79 4. Potential theory on complex projective space: application to characterization of pluripolar sets and growth of analytic varieties, Illinois J. Math. 28, N° 1 (1984), 103-119 Molzon, R.E., Shiffman, B., Sibony, N. 1. Average growth estimates for hyperplane sections of entire analytic sets, Math. Ann. 257 (1981), N° 1,43-59 MOrZakov, V.V. 1. Convolution equations in spaces of functions that are holomorphic in convex domains and on convex compacta in ([:n (Russian), Mat. Zametki 16 (1974), 431-440. English translation: Math. Notes 16 (1974), N° 3, 846-851 (1975) Michiwaki, Y. 1. Several complex variables and Picard's theorem, Sci. Rep. Tokyo Kysiku Daigaku sect. A 5 (1955),77-81 Motzkin, T.S., Schoenberg, l.1. 1. On lineal entire functions of n complex variables, Proc. Amer. Math. Soc. 3 (1952), 517-526 Muhtarov, A.S. 1. The growth of entire functions of two complex variables (Russian), Akad. Nauk. Azerbaid• zan, SSR., Ook!. 27 (1971), N° 3,6-9 2. The characterization of the growth of functions (Russian), Akad. Nauk. Azerbaidzan, SSR, Ook!. 28 (1972), N° 4, 13-15 Napalkov, V.V. 1. Subspaces of entire functions of exponential type that are translation invariant (Russian), Sibirisk. Mat. Z. 14 (1973), 427-426, 463. English translation: Siberian Math. 1. 14 (1973), 294-300 Newman, OJ., Shapiro, H.S. 1. Fischer spaces of entire functions. Entire Functions and Related Parts of Analysis (Proc. Sympos. Pure Math., La Jolla, Calif., 1966), 340-349, Amer. Math. Soc. Providence, R.I. (1968) Nigram, H.N. 1. Use of the generalized Laplace transform to integral functions of several complex variables, Riv. Mat. Univ. Parma, (2), 7 (1966), 137-144 2. Some uses of the basic properties of Meijer transform to integral funct'ions of two complex variables, Riv. Mat. Univ. Parma, (2), 7 (1966), 193-202 3. On "Borel-Laplace" transforms and integral functions of two complex variables, Istambul Univ. Feu. Fak. Mecm. Ser. A33 (1968),51-62 (1971) Nikol'skii, S.M. 1. Some inequalities for entire functions of finite degree of several variables and their appli• cation (Russian), Ooklady Akad. Nauk. SSSR (N.S.), 76 (1951), 785-788 Nishino, T. 1. Sur les valeurs exceptionnelles au sens de Picard d'une fonction entiere de deux variables, J. Math. Kyoto Univ. 2 (1962-1963),365-372 264 Bibliography 2. Nouvelles recherches sur les fonctions entieres de plusieurs variables complexes I, J. Math. Kyoto Univ. 8 (1968),49-100 3. Nouvelles recherches sur les fonctions entiere de plusieurs variables complexes. II. Fonctions entieres qui se n\duisent a celles d'une variable, J. Math. Kyoto, Univ. 9 (1969), 221-274 4. Nouvelles recherches sur les fonctions entieres de plusieurs variables complexes. III. Sur quel• ques proprietes topologiques des surfaces premieres, J. Math. Kyoto Univ. 10 (1970), 245-271 5. Nouvelles recherches sur les fonctions entieres de plusieurs variables complexes. IV. Types de surface premiere, 1. Math. Kyoto Univ. 13 (1973), 217-272 6. Nouvelles recherches sur les fonctions entieres de plusieurs variables complexes. V. Fonctions qui se reduisent aux polynomes, J. Math. Kyoto Univ. 15 (1975), N° 3,527-553 Nishino, T., Yoshioka, T. 1. Sur I'iteration des transformations rationnelles entiere de I'espace de deux variables com• plexes, C.R. Acad. Sci. Paris, 240 (1965), 3835-3837 Noverraz, Ph. 1. Comparaison d'indicatrices de croissance pour des fonctions plurisousharmoniques ou entieres d'ordre fini, J. Analyse Math. Jerusalem 12 (1964), 409-418 2. Fonctions entieres ou plurisousharmoniques de type exponentiel, Ann. Soc. Sci. Bruxelles, Ser. 1,75 (1961), 113-122 Okada, M. 1. Un theoreme de Bezout transcend ant sur 8. The growth of plurisubharmonic functions and the distribution of values of entire functions of several variables (Russian), Dokl. Akad. Nauk. SSSR, 179 (1968), 290-292 9. The analog of the canonical product for entire functions of several complex variables (Russian), Trudy Moskov. Mat. Obsc. 18 (1968),105-146 10. Characterizations of the distribution of seros of entire functions of several variables (Rus• sian), Teor. Finkeii Funkcional. Anal. i. Prilozen typ 12 (1970),111-116 11. The completeness of function system e'd.x) and real uniqueness sets of entire functions of several variables (Russian), Funkcional Anal. i. Prilozen, 5 (1971), N° 4, 86 12. Certain questions of the distribution of the zeros of entire functions of several variables (Russian), Mat. Sb. (N.S.), 87, (129) (1972), 351-368 13. Real sets of uniqueness for entire functions of several variables and the completeness of systems of functions e' Sire, 1. 1. Sur les fonctions de deux variables d'ordre apparent total fini, Rend. Circ. Palermo 31 (1911), 1-91 Siu, Y.T. 1. Analyticity of sets associated to Lelong numbers and the extension of closed positive currents, Invent. Math. 27 (1974), 53-156 Skoda, H. 1. Solution it croissance du second probleme de Cousin dans Parts of Analysis (Proc. Sympos. Pure Math .. La Jolla, Calif., 1966), 392-430, Amer. Math. Soc. Providence, R.I. 1968 6. About the value distribution of holomorphic maps into the projective space, Acta Math., 123 (1969),83-114 7. Value distribution of holomorphic maps. Several Complex Variables, I (Proc. Conf. Univ. of Maryland, College Park, Md., 1970, 165-190, Springer, Berlin (1970) 8. A Bezout estimate for complete intersections, Ann. of Math. (2), 96 (1972), 361-401 9. Holomorphic functions of finite order on several complex variables. Conference Board of the Mathematical Sciences, Regional Conferences Series in Mathematics, N° 21, American Math• ematical Society, Providence, R.I. (1974) Strelic, S.l. I 1. The Wiman-Valiron theorem for entire functions of several variables, Dokl. Akad. Nauk. SSSR, 134 (1960), 286-288 (Russian), English translation: Soviet Math. Dokl. 1 (1961), 1075- 1077 2. Generalization to entire functions of several complex variables of the theorem of Wiman and Valiron (Russian), Litovsk. Mat. Sb. 1 (1961). N° 1-2,327-354 3. Relations for the derivatives of an entire transcendental function of several variables at points of maximum modulus (Russian), Dokl. Akad. Nauk. SSR, 145 (1962), 737-740 4. On the maximum modulus of analytic functions of several variables (Russian), Mat. Sb. (N.S.), 57, (99) (1962), 281, 296 5. Some questions of the growth and existence of entire transcendental solutions of partial differential equations (Russian), Litovsk. Mat. Sb. 2 (1962), N° 1,167-178 6. Some properties of the maximum modulus of analytic functions of several variables (Rus• sian), Litovsk. Mat. Sb.Z. (1962), N° 1, 153-166 7. The theorem of Wiman and Valiron for entire functions of several complex variables (Russian), Mat. Sb. (N.S.), 58, (100) (1962), 47-64 8. The growth of entire solutions of partial differential equations (Russian), Mat. Sb. (N.S.), 61, (103) (1963), 257-271 9. Behavior of an entire transcendental function of several complex variables for large values of its modulus (Russian), Litovsk. Mat. Sb. 4 (1964), 357-408 Suzuki, M. 1. Proprietes topologiques des polynomes de deux variables complexes et automorphismes algebriques de l'espace Waldschmidt, M. 1. Nombres transcend ants et groupes algebriques, Asterisque 69-70, Soc. Math. France (1969) Wang, S.P. 1. On difference equations of entire functions, Chinese 1. Math. 2 (1974), N° 2, 291-306 Wiegerinck,l. 1. Growth properties of functions of Paley-Wiener class on (Cn, Indagationes Math. 46, N° 1 (1984) 2. Paley-Wiener functions with prescribed indicator, Thesis, Univ. Amsterdam Winiarski, T.D. 1. Approximation and interpolation methods in the theory of entire functions of several variables (Polish and Russiam summaries). Proceedings of the Fifth Conference on Analytic Functions (Univ. Mariae Curie-Sklodowska, Lublin, 1970). Ann. Univ. Mariae Curie-Sklo• dowska, Sect. A, 22-24,1968-1970,189, 191 (1972) 2. Applications of approximation and interpolation methods to the examination of entire functions of n complex variables, Ann. Polon. Math. 28 (1973), 97-121 Wirtinger, W. 1. Eine Determinatenidentitat und ihre Anwendung auf analystische Gebilde, Monatsh.Math. und Pysik 441 (1936), 343-365 WU,H. 1. Normal families of holomorphic mappings. Acta. Math. 119 (1967), 193-233 2. An n-dimensional extension of Picard's theorem, Bull. Amer. Math. Soc. 75 (1969), 1357- 1361 Yamaguchi, H. 1. Sur une uniformite des surfaces constantes d'une fonction entiere de deux variables com• plexes, 1. Math. Kyoto Univ. 13 (1973), 417-433 2. Sur Ie mouvement des constantes de Robin, 1. Math. Kyoto Univ. 15 (1975), 53-71 3. Parabolicite d'une fonction entiere, 1. Math. Kyoto Univ. 16 (1976), N° 1,71-92 Index Algebraic dependence 160 -- decomposable 31 - independence 160 - pull back 120 - integer 155 Fourier-Borel transform 178/204 - number 155 'iJ -support 186 -- size 156 - variety 7 Grassmannian 143 Analytic functional 177 Genus 64 - variety 46 Harmonic function 230 Bounded family of polynomials 82 Hartog's Lemma 22 Holomorphic function 2 Carrier 177 Cauchy-Fantappie Formula 41 Indicator of growth function, circled 21 Compactly contained in 106 --- Cousin data 63 Complete intersection 47 --- positive current 37 - left stability 126 --- projective 179 Complex dimension 47 --- radial 21 - homogeneous function 5 --- with respect to one variable 11 - submanifold 46 Inverse Function Theorem for Plurisubhar• Convergence exponent 64 monic Functions 34, 240 Convolution operator 207 Cousin data 59 Laplace transform, generalized 186 - area of 62 -- projective 183 - current of integration 62 Lelong number 37 - multiplicity 60 Linearly separates 195 Current 34 - closed 37 Maximum Principle 234 - continuous of order zero 36 Minimal growth class 134 - dominates 36 - positive 34 Order 8 -- degree 34 - proximate 14 - push forward 120 - conjugate 205 - strong proximate 16 Denominator 156 - total 11 Division Theorem 208 - with respect to one variable, 12 Entire function 2 Pluriharmonic function 231 Extension, finite type 155 Pluripolar set 24, 234 - simple 155 Plurisubharmonic function 3, 230 -- locally 237 Form modulus 35 Polar set 234 - norm 35 Polynomial domination 159 - positive 30 - size 159 270 Index Positively homogeneous function 5 Supporting function 178, 185 Pseudo-algebraic 136 - hyperplane 197 Transcendence basis 160 Regular growth 96 - degree 160 - system 32 Transcendental number 155 Regularization 19 Type 8, 14 - maximal 8 - minimal 8 Slowly increasing function 14,79 - normal 8 Subadditive function 5 Subharmonic function 3, 230 Weierstrass Preparation Theorem 47 -- locally 237 - pseudo-polynomial 47 Grundlehren der mathematischen Wissenschaften A Series ojComprehensive Studies in Mathematics A Selection 190. Faith: Algebra: Rings, Modules, and Categories I 191. Faith: Algebra II, Ring Theory 192. Mal'cev: Algebraic Systems 193. P6lya/Szego: Problems and Theorems in Analysis I 194. Igusa: Theta Functions 195. Berberian: Baer*-Rings 196. Athreya/Ney: Branching Processes 197. Benz: Vorlesungen tiber Geometrie der Algebren 198. Gaal: Linear Analysis and Representation Theory 199. Nitsche: Vorlesungen tiber Minimalflachen 200. Dold: Lectures on Algebraic Topology 201. Beck: Continuous Flows in the Plane 202. Schmetterer: Introduction to Mathematical Statistics 203. Schoeneberg: Elliptic Modular Functions 204. Popov: Hyperstability of Control Systems 205. Nikol'skiI: Approximation of Functions of Several Variables and Imbedding Theorems 206. Andre: Homologie des Algebres Commutatives 207. Donoghue: Monotone Matrix Functions and Analytic Continuation 208. Lacey: The Isometric Theory of Classical Banach Spaces 209. Ringel: Map Color Theorem 210. Gihman/Skorohod: The Theory of Stochastic Processes I 211. Comfort/Negrepontis: The Theory of Ultrafilters 212. Switzer: Algebraic Topology - Homotopy and Homology 215. Schaefer: Banach Lattices and Positive Operators 217. Stenstrom: Rings of Quotients 218. Gihman/Skorohod: The Theory of Stochastic Processes II 219. DuvantiLions: Inequalities in Mechanics and Physics 220. Kirillov: Elements of the Theory of Representations 221. Mumford: Algebraic Geometry I: Complex Projective Varieties 222. Lang: Introduction to Modular Forms 223. Bergh/Lofstrom: Interpolation Spaces. An Introduction 224. Gilbarg/Trudinger: Elliptic Partial Differential Equations of Second Order 225. Schtitte: Proof Theory 226. Karoubi: K-Theory. An Introduction 227. Grauert/Remmert: Theorie der Steinschen Riiume 228. Segal/Kunze: Integrals and Operators 229. Hasse: Number Theory 230. Klingenberg: Lectures on Closed Geodesics 231. Lang: Elliptic Curves: Diophantine Analysis 232. GihmaniSkorohod: The Theory of Stochastic Processes III 233. StroockiVaradhan: Multidimensional Diffusion Processes 234. Aigner: Combinatorial Theory 235. Dynkin/Yushkevich: Controlled Markov Processes 236. Grauert/Remmert: Theory of Stein Spaces 237. Kothe: Topological Vector Spaces II 238. Graham/McGehee: Essays in Commutative Harmonic Analysis 239. Elliott: Probabilistic Number Theory I 240. Elliott: Probabilistic Number Theory II 241. Rudin: Function Theory in the Unit Ball of cn 242. Huppert/Blackburn: Finite Groups II 243. Huppert/Blackburn: Finite Groups III 244. Kubert/Lang: Modular Units 245. Cornfeld/FominlSinai: Ergodic Theory 246. NaimarkiStem: Theory of Group Representations 247. Suzuki: Group Theory I 248. Suzuki: Group Theory II 249. Chung: Lectures from Markov Processes to Brownian Motion 250. Amold: Geometrical Methods in the Theory of Ordinary Differential Equations 251. Chow/Hale: Methods of Bifurcation Theory 252. Aubin: Nonlinear Analysis on Manifolds. Monge-Ampere Equations 253. Dwork: Lectures on p-adic Differential Equations 254. Freitag: Siegelsche Modulfunktionen 255. Lang: Complex Multiplication 256. Hormander: The Analysis of Linear Partial Differential Operators I 257. Hormander: The Analysis of Linear Partial Differential Operators II 258. Smoller: Shock Waves and Reaction-Diffusion Equations 259. Duren: Univalent Functions 260. FreidlinlWentzell: Random Perturbations of Dynamical Systems 261. Bosch/GUntzer/Remmert: Non Archimedian Analysis - A Systematic Approach to Rigid Geometry 262. Doob: Classical Potential Theory and Its Probabilistic Counterpart 263. Krasnosel'skil'/Zabreiko: Geometrical Methods of Nonlinear Analysis 264. AubiniCellina: Differential Inclusions 265. Grauert/Remmert: Coherent Analytic Sheaves 266. de Rham: Differentiable Manifolds 267. Arbarello/Comalba/Griffiths/Harris: Geometry of Algebraic Curves, Vo1.I 268. Arbarello/Comalba/Griffiths/Harris: Geometry of Algebraic Curves, Vol. II 269. Schapira: Microdifferential Systems in the Complex Domain 270. Scharlau: Quadratic and Hermitian Forms 271. Ellis: Entropy, Large Deviations, and Statistical Mechanics 272. Elliott: Arithmetic Functions and Integer Products 274. Hormander: The Analysis of Linear Partial Differential Operators III 275. Hormander: The Analysis of Linear Partial Differential Operators IV 276. Liggett: Interacting Particle Systems 277. FultoniLang: Riemann-Roch Algebra 278. Barr/Wells: Toposes, Triples and Theories 279. Bishop/Bridges: Constructive Analysis 281. Chandrasekharan: Elliptic Functions Springer-Verlag Berlin Heidelberg New York Tokyo